The grids provided and used to date in the petroleum sphere are of three types: entirely structured, totally unstructured or hybrid, i.e. a mixture of these two grid types.
Structured grids are grids whose topology is fixed: each inner vertex is incident to a fixed number of cells and each cell is delimited by a fixed number of faces and edges. Cartesian grids (FIG. 2), which are widely used in reservoir simulation, CPG (Corner-Point-Geometry) type grids, described for example in patent FR-2,747,490 (U.S. Pat. No. 5,844,564) filed by the applicant, and grids of circular radial type (FIG. 1) allowing the drainage area of the wells to be modelled, can be mentioned for example.
Unstructured grids have a completely arbitrary topology: a vertex of the grid can belong to any number of cells and each cell can have any number of edges or faces. The topological data therefore have to be permanently stored to explicitly know the neighbours of each node. The memory cost involved by the use of an unstructured grid can therefore become rapidly very penalizing. However, they allow to describe the geometry around wells and to represent complex geologic zones. The grids of PErpendicular Blssector (PEBI) or Voronoï type provided in the following document can for example be mentioned: Z. E. Heinemann, G. F. Heinemann and B. M. Tranta, “Modelling heavily faulted reservoirs”, Proceedings of SPE Annual Technical Conferences, pp. 9-19, New Orleans, La., September 1998, SPE. Structured grids have already shown their limits: their structured nature facilitates their use and implementation, but they therefore involve a rigidity that does not allow all the geometric complexities of geology to be represented. Unstructured grids are more flexible and they have allowed to obtain promising results but they still are 2.5D grids, i.e. the 3rd dimension is obtained only by vertical projection of the 2D result, and their lack of structure makes them more difficult to exploit.
In order to combine the advantages of both approaches, structured and unstructured, while limiting the drawbacks thereof, another type of grid has been considered: the hybrid grid. It is a combination of different grid types and it allows to make the most of their advantages, while trying to limit the drawbacks thereof. A hybrid local refinement method is proposed in: O. A. Pedrosa and K. Aziz, “Use of hybrid grid in reservoir simulation”, Proceedings of SPE Middle East Oil Technical Conference, pp. 99-112, Bahrain, March 1985. This method consists in modelling a radial flow geometry around a well in a Cartesian type reservoir grid. The junction between the cells of the reservoir and of the well is then achieved using hexahedral type elements. However, the vertical trajectory followed by the centre of the well must necessarily be located on a vertical line of vertices of the Cartesian reservoir grid.
To widen the field of application of this method, in order to take account of the vertical and horizontal wells and of the faults in a Cartesian type reservoir grid, a new hybrid local refinement method has been proposed in: S. Kocberber, “An automatic, unstructured control volume generation system for geologically complex reservoirs”, Proceedings of the 14th SPE Symposium on Reservoir Simulation, pp. 241-252, Dallas, June 1997. This method consists in joining the reservoir grid and the well grid, or the reservoir grid blocks to the fault edges, by pyramidal) prismatic, hexahedral or tetrahedral type elements. However, the use of pyramidal or tetrahedral cells does not allow a cell-centered finite volume type method to be used.
Patents FR-2,802,324 and FR-2,801,710 filed by the applicant describe another type of hybrid model allowing to take into account, in 2D and 2.5D, the complex geometry of reservoirs and the radial directions of flow in the areas around wells. This hybrid model allows very precise simulation of the radial nature of the flows around wells by means of a cell-centered finite volume type method. It is structured nearly everywhere, which facilitates its use. The complexity inherent in the lack of structure is introduced only where it is strictly necessary, i.e. in the transition zones of reduced size. Calculations are fast and taking account of the directions of flow through the geometry of the wells increases their accuracy. Although this 2.5D hybrid grid has allowed to take a good step forward in reservoir simulation in complex geometries, the fact remains that this solution does not allow to obtain an entirely realistic simulation when the physical phenomena modelled are really 3D. This is the case, for example, for a local simulation around a well.
Furthermore, these hybrid grid construction techniques require creating a cavity between the reservoir grid and the well grid. S. Balaven-Clermidy describes, in “Génération de maillages hybrides pour la simulation des réservoirs pétroliers” (thesis, Ecole des Mines, Paris, December 2001), various methods for defining a cavity between the well grid and the reservoir grid: the minimum size cavity (by simple deactivation of the cells of the reservoir grid overlapping the well grid), the cavity obtained by expansion and the cavity referred to as Gabriel cavity. However, none of these methods is really satisfactory: the space created by the cavity does not allow the transition grid to keep an intermediate cell size between the well grid cells and the reservoir grid cells.
Furthermore, patent application EP/05-291,047,8 filed by the applicant describes another hybrid type method allowing to take into account, in 2D, 2.5D and 3D, the complex geometry of reservoirs and the radial directions of flow in the areas around wells it consists in generating entirely automatically a cavity of minimum size while allowing the transition grid to keep an intermediate cell size between the size of the well grid cells and the size of the reservoir grid cells. This method also allows to construct a transition grid meeting the constraints of the numerical scheme used for simulation. This method provides optimization techniques consisting in providing a posteriori improvement of the hybrid grid, to define a perfectly admissible transition grid in the sense of the numerical scheme selected. This hybrid approach allows to connect a non-uniform Cartesian type reservoir grid to a circular radial type well grid.
However, reservoir modelling by a Cartesian grid is not sufficient to take account of all the geologic complexity thereof. It is therefore necessary to use Corner Point Geometry (CPG) type structured grids to represent them. Generally, CPG grids have quadrilateral faces whose vertices are neither cospherical nor coplanar. The edges of these grids are even, in most cases, non-Delaunay admissible, i.e. the diametral spheres of some edges are non-empty. Now, the method described above cannot manage this type of grid suitably. Current methods allowing hybrid grids to be generated are therefore no longer applicable in the precise case of CPG grids.
The technique described in patent FR-2,891,383 allows to overcome this problem. This method allows to construct entirely automatically conformal transition grids when the reservoir is described by a CPG type grid. It consists in locally deforming a CPG type grid into a non-uniform Cartesian grid. These local grid cell deformations correspond to the change from a reference frame referred to as “CPG” frame to a reference frame referred to as “Cartesian” frame defined by the deformation. These deformations are then quantified and applied to the structured grids so as to shift to the “Cartesian” frame. A hybrid grid is then generated in the “Cartesian” frame from the two grids thus deformed. Finally, this hybrid grid is deformed to return to the “CPG” frame, prior to improving the grid quality by optimizing it under quality control in the sense of the numerical scheme.
However, according to this technique, the stage of local deformation of the CPG type grid to a non-uniform Cartesian grid also modifies the radial grid. In terms of industrial application, this technique is thus limited to cases where the induced deformations remain low, i.e. when the petroleum reservoir is not very complex.
The object of the invention thus relates to an alternative method for evaluating fluid flows in a heterogeneous medium. The method comprises constructing a hybrid grid from a CPG type grid and a radial grid. This method overcomes the aforementioned problems by deforming the CPG type grid into a non-uniform Cartesian grid and by defining, then by correcting a deformation function so that the radial grid in the Cartesian frame keeps the geometrical characteristics necessary for construction of a transition grid.